# Look for and Express Regularity in Repeated Reasoning

• Multiplication tables
• Divisibility rules
• Short-division, and
• Prime factorization.
These are the tools of success in mastering fractions and then algebraic operations. These skills are best derived by patterns and then practiced using these patterns. To make the observation of regularity, pattern, repeated reasoning, and structure possible, both teachers and students need to pose a series of questions such as:
• Is there a process—operation, property, condition that I am repeating?
• Can I use this repeated idea to create a pattern and then to develop a conjecture?
• Can I make a generalization of the conjecture?
• Can I express this abstractly based on this repeated reasoning?
• Does this abstraction result into a procedure, method, or strategy?
• Are there exceptions to this generalization/abstraction?
• Can I make this method, procedure more efficient and elegant (this is like editing a piece of writing)
Teachers can ask questions such as:
• You claim that you observed a pattern, how do you know it is a pattern?
• Can you describe your pattern, conjecture, or method?
• Explain why your method makes sense?
• Can you describe your method in formal mathematics terms and reasoning?
• How would you explain why it works?
• Have you checked whether the answer seem reasonable?
• What have you learned about ….?
• How would this work with other numbers or situations? Does it work all the time? How do you know?
• What do you notice when …?
Mathematics is about constructing and solving problems, and interesting problems must be the focus of students’ mathematical experiences; they should be the content of daily mathematics lessons. Problems should be accessible to children, yet they should be modestly challenging and at times creatively frustrating. These kinds of problems should be the focus when students and their teachers are engaged in sharing their thinking process. These thinking processes— having ideas, not having ideas, seeing relationships, discovering patterns, making conjectures, constructing examples and counterexamples, devising arguments, and critiquing each other’s work are the back-bone of the standards of mathematics instruction. Standards of mathematics practice—techniques and methods, when routinely practiced, will implement the content advocated in the CCSSM. Arising naturally out of practice, content will not be in isolated from; it will be an organic outgrowth of the problem background. When a concept and procedure are learned, they should be practiced till they convert into skills and tools for doing meaningful mathematics. Doing mathematics is different than knowing mathematics. However, practice is a good reinforcement tool if students have learned the content first. The Standards of Mathematics Practices (SMP) aim to develop mathematics content, the mathematical way of thinking, and mastery of the content with deeper understanding. As students work on solving problems, they are guided to observe regularity and symmetry, discover relationships, and make conjectures. They are asked to evaluate the reasonableness of their intermediate results and arrive at mathematical concepts. They articulate these results and then prove or disprove their results using formal reasoning and their prior knowledge of related concepts. The success of this evaluation process is dependent on several skills that can be developed with the following activities:
• Encourage students to share multiple methods for solving the same problem and then request that students use someone else’s method to solve a similar problem.
• After sharing and naming multiple methods for a problem, tweak the parameters and conditions of the problem and ask students “Which method would you use to solve the problem if it were like this?”
• Playfully and explicitly add an element of time to students’ problem solving, once they have had experience developing methods for a particularly problem structure. For example, if students have painstakingly drawn pictures and calculated the sum of the interior angles of rectangles, rhombuses, pentagons, and hexagons, and shared their different methods for doing so, ask them to think about ways they could get more efficient. Then ask them to make a bid for how many polygons they could find interior angle sums for in 5 minutes. Pass out the requested number of puzzles to each team, and start the clock. While you’ll have to pay attention to group dynamics and support students to be good team members even under pressure, you’ll also hear good ideas about mathematical shortcuts that didn’t come up when students didn’t have any reason to need more efficient methods.
Another purpose of this standard is to make sense of the formulas we use by finding a pattern or relationship in numbers generated during an exploration of the topic. It is to understand that there are purposeful connections between procedures and concepts or between the manipulation of tools and a procedure. For example, students can explore the area of a triangle by building rectangles, drawing the diagonal, and realizing that the area of a triangle is half of the area of a rectangle formed with the same base and same height—thus the formula is developed and then relate to the parallelograms of the same base and height. It is important to design lessons where patterns are revealed and lessons that encourage students to make generalizations. Some of these tasks involve students working with prior knowledge in a non-routine way. The learning that takes place through these lessons is incredible. Using the best-designed lessons is not the only necessary component in developing this practice in students, so be sure to facilitate conversations that include comments and questions such as:
• Notice repeated calculations and reasoning
• What generalizations can you make?
• Look for general methods and shortcuts
• What mathematical consistencies do you notice?
• How would you prove that …?
• What would happen if …?
• What predictions or generalizations can this pattern support?
• How would this strategy work with another number? Does it work all the time? How do you know?
• Maintaining over sight of process while looking at details evaluate reasonableness of results
• Finding new structures/methods, generalizing
• Can you find a shortcut to solve the problem? How would your shortcut make the problem easier?